A median of a triangle is a segment joining a vertex to the midpoint of the opposite side....

A median of a triangle is a segment joining a vertex to the midpoint of the opposite side. Using vectors, show that the point in which two medians intersect cuts them both into two segments such that the lengths of the subsegments are in the ratio 2 to 1. See Figure.

(Suggestion: The vectors and are linearly dependent, so for some scalar k1. Similarly, for some other constant k2. Verify that

Use this to show that

and use linear independence.)

Figure

Intersecting medians of a triangle